Hedging cost 就是你的衍生品价格。为什么要完备市场？比如说，你的衍生品的风险来自相对独立的三个随机方向（设想是三个线性无关的向量），假如市场中有trade基于这些风险的vanilla products，you can use linear combinations of these products to form a portfolio that replicates the risk of your derivative product. 也就是说，这些vanilla products span了你的风险空间，你可以做出任何基于这些风险的衍生品，并且能够fully and dynamically hedge the risk by the replicating portfolio. 因为你hedge掉了所有的风险，所以的总资产是无风险的，所以你的总资产的收益必须在risk free rate，所以你才没有了arbitrage。

总之，你的衍生品价格 must equal to the cost of your hedging portfolio，假如你有一个方向的风险无法replicate，那你的衍生品价格无法确定，换句话说就是有arbitrage。

It is for pricing purpose under the no arbitrage pricing framework

Basically if the market is complete it indicates no arbitrage opportunity and uniqueness of risk neutral measure. The advantage of this assumption is it can reduce the complexity of modeling. If the market is complete, your modeling can be limited in the scope of the fundamental assets and the derivatives. In BS's case it is option, underlying equity and cash instrument. As long as you can replicate the option with the stock and cash instrument, you find the option's price. But if the market is incomplete, because people can not hedge all the risk of the derivative, the remaining "price" of the unhedged risk actually depends on many exogenous factors like people's risk preference. That will make the model fall back to equilibrium modeling in economics. In finance, especially in mathematical finance, people usually do in the no arbitrage way because that will enable many math convenience and beauty of the model.

Complete means the number of your fundamental assets is the same as the number of independent risk source e.g. brownian motion. I don't know why people call it "complete" but from my perspective maybe it is because in that market people can completely eliminate the risk via hedging.  

Thank you.